Near Tool, get distance in degrees instead of meters

Hello everybody,

I have two point shapefiles and I need to calculate the distance in meters from all points in one file to the nearest point in the second file. It works fine, but I get the distance in degrees.

I googled the problem and it seems that the data needed to be projected to be able to get the distance in meters. I used the "Define Projection" and the "Project" tools to make sure my data is projected. Then I exported both features to shapefiles. In the properties of the two shapefiles one can read:

Projected Coordinate System: SWEREF99_TM
Projection: Transverse_Mercator
False_Easting: 500000,00000000
False_Northing: 0,00000000
Central_Meridian: 15,00000000
Scale_Factor: 0,99960000
Latitude_Of_Origin: 0,00000000
Linear Unit: Meter

Geographic Coordinate System: GCS_SWEREF99
Datum: D_SWEREF99
Prime Meridian: Greenwich
Angular Unit: Degree

When I run the "Near" tool I still get the distance in degrees. Is there anything I am missing?

Thank you,

Oana